- DERIVATIVE ESTIMATES AND SOME CONJECTURES 153
1
R(O'(u))::::; R-1 (0'(0)) + (1-28)u·
From this we may obtain
COROLLARY 20.24. For a 3-dimensional nonfiat steady gradient Ricci
soliton with Rm 2'.: 0 which is /'i,-noncollapsed at all scales we have
c
R(x) <---- 1+d(x,p)
for some C < oo.
Finally, by (20.52) this implies Re (V' f, V' f) :S (l+d~,p)) 2 for some C <
00.4.3. Conjectures on the classification of ancient solutions in
low dimensions.
Perelman conjectures (see §11.9 of [152]) that there is only one (up to
scaling) noncompact 3-dimensional /'i,-solution with positive sectional curva-
ture: the rotationally symmetric gradient steady soliton of Bryant. Recall
that Corollary 20.11 says that the collection of 3-dimensional /'i,-solutions
is compact modulo scaling. More strongly, Perelman's conjecture identifies
all noncompact 3-dimensional /'i,-solutions as either 52 x IR.^1 , its Z2-quotient,
or the Bryant soliton. Perelman states in §11.9 of [152] that he can prove
uniqueness in the subclass of those 3-dimensional gradient steady solitons
which are /'i,-solutions. Note that, by solving an ODE, it is not difficult to
prove this in the rotationally symmetric case.
PROBLEM 20.25. Prove that a rotationally symmetric noncompact 3-
dimensional /'i,-solution with positive sectional curvature must be a constant
multiple of the Bryant soliton.
Slightly more generally, we venture to conjecture the following.^22
OPTIMISTIC CONJECTURE 20.26. The only 3-dimensional /'i,-Solutions
with positive sectional curvature are
(a) 53 /r with constant sectional curvature,
(b) the Bryant steady soliton, and
( c) Perelman's /'i,-solution on either 53 or IR.P^3.
In dimension 2 we have the following.
OPTIMISTIC CONJECTURE 20.27. Any 2-dimensional ancient solution
with bounded curvature is either
(1) 52 , IR.P^2 , or a quotient ofIR.^2 with a metric of constant nonnegative
curvature,
(2) the cigar steady soliton, or
(3) the King-Rosenau ancient solution on 52 or IR.P^2.
(^22) We owe this numbering system to The Jackson 5.